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Responding to Infectious Diseases Using Game
Theory
Table of Contents:
Abstract:
Impact:
Potential Applications:
Question/Hypothesis:
Background Research:
Game Theory:
Epidemic Simulation:
Virus Model:
Strategies Investigated:
Public Health Strategies:
Viral Strategies:
Payoff Calculations:
Viral Payoff:
Public Health Payoff:
Data and Results:
Simulation Results:
The Sniffles: An Iterative Dominance Game:
Spattergroit: A Prisoner’s Dilemma Game:
Conclusion:
Future Research:
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Abstract
The research analyzed interactions between human public health organizations and
infectious diseases (specifically viruses) in order to attempt to investigate game
theory’s applications to conflicts and interactions between humans and infectious
diseases. The research set up games to mimic the interaction between humans and
infectious disease by treating both as rational players with discreet payoffs. The
procedure consisted of running simulations utilizing the Framework for
Reconstructing Epidemiological Dynamics (FRED) from the University of Pittsburgh
and varying parameters to set up strategies for the virus and public health
organizations and determine the resulting payoffs. The data retrieved consisted
primarily of a pair of games, one solvable through iterative dominance and the other
an example of the Prisoner’s Dilemma. The iterative dominance game provided an
example of a way in which game theory can provide a route to solutions previously
not considered by treating the virus as a rational player that will make the best
choice for its reproductive success. In addition, it showed the cycle that is
exemplified in an extreme form by symbiotic enterobacteriaceae that adopt an
asymptomatic and even beneficial strategy while humans choose not to attempt to
prevent their further spread. The Prisoner’s Dilemma game suggested that game
theory could possibly lead to cycles of cooperation with a disease that could be
either continued indefinitely or used to weaken a virus until a sudden defection
from cooperation by health organizations could destroy it.
Impact
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Better analysis of public health strategies by treating viruses as rational
players
Mitigation of Viral symptoms through a Nash Equilibrium
Opportunities for cooperation with and defection against a virus in order to
advance public health goals
Potential Applications
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Quarantines and Bird Flu
Travel restrictions and Ebola
Measles and vaccination policy
Predicting the prevalence of Flu strains for vaccination
Question/Hypothesis
Game Theory can be applied among humans to fields from economics to
international relations. Likewise, Evolutionary Game Theory can be applied to
explain many biological behaviors that otherwise are enigmatic. However, there
have been relatively few studies attempting to combine the two. By studying
interactions between human public health organizations and diseases through Game
Theory, this research investigates a new field of potential interactions.
Background Research:
Game Theory
Game Theory was first developed by John von Neumann in his studies of two-player
zero sum games, or games in which any gain by one player must result in a loss by
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the other player. Since then, it has been expanded to include cooperative games (in
which a good outcome for one player is not a bad outcome for another) and games
with three or more players. The central concept of game theory is the Nash
Equilibrium, an outcome for which neither player has an incentive to unilaterally
deviate (switch strategies by themselves) from their chosen strategies. If both
players play perfectly (i.e. rationally), the two strategies will result in the Nash
Equilibrium. In order to find Nash Equilibria, Game Theorists use a payoff matrix,
which details the rewards for each player given that each player chooses a certain
strategy. The payoff matrix for a Prisoner’s Dilemma, an oft-studied game, is shown
below.
Player A
Player B
Cooperate
Defect
Cooperate
2 , 2
0 , 4
Defect
4 , 0
1 , 1
In the Prisoner’s Dilemma, each player chooses between one of two strategies,
either to cooperate or to defect. The payoff is represented as a pair of values, the
first generally for the player on the left of the matrix and the second for the player
above the matrix. The payoff for a certain pair of strategies is given by the pair of
values in the cell on the same row as Player A’s strategy and the same column as
Player B’s strategy. Note that at when both players choose Defect, neither player has
an incentive to unilaterally deviate to Cooperate from their current payoff of one to
a payoff of zero, despite the fact that if each chose Cooperate, each player would
have a better payoff. Thus, (Defect, Defect) is a Nash Equilibrium, while (Cooperate,
Cooperate) is not a Nash Equilibrium, because at (Cooperate, Cooperate), each
player has an incentive to deviate to Defect and gain a payoff of four. Also, note that
when determining the Nash Equilibria studied in this research, the magnitude of
each value does not matter save for with respect to the other values. If (Cooperate,
Cooperate) produced payoffs of (3, 3), the game would remain the same, because
the change between (2, 2) and (3, 3) did not cause the payoffs of (Cooperate,
Cooperate) to be greater or less than any other pair of strategies that (Cooperate,
Cooperate) was not greater or less than before the change in payoffs.
Epidemic Simulation
The Framework for Reconstructing Epidemiological Dynamics (FRED) is a
simulation created by the University of Pittsburgh Public Health Dynamics
Laboratory in collaboration with the Pittsburgh Supercomputing Center and the
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School of Computer Science at Carnegie Mellon University. FRED is designed to
simulate the spread of infectious diseases and the interplay between public health
organizations, viral evolution, and personal health behavior. FRED uses agents
representing the population of a county to simulate the spread of infectious
diseases. Agent statistics are determined by census data, and each individual agent
actively moves through environments such as their neighborhood, workplace, or
school. Each agent is individually simulated, as are the infection events resulting
from the interaction between an infected and an uninfected agent. The simulation
allows for parameters changing the behavior of agents, changing the infectivity (in
both asymptomatic and symptomatic stages), dormancy time, and other traits of the
virus, and changing the responses by public health organizations to mitigate the
virus’s spread.
Virus Model
The simulated virus followed the sequential SEiIR model of infection in each agent:
 Susceptible
 Exposed
 infectious (asymptomatic)
 Infectious (symptomatic)
 Recovered
An important variable in the model is the asymptomatic infectivity factor, which is
the ratio of the virus infectivity in the asymptomatic phase to that in the
symptomatic phase.
Strategies Investigated
Public Health Strategies
Two strategies were investigated for the public health organization:
(A) A Laissez-faire policy where each symptomatic agent may choose whether to
stay home from work on a given day with a likelihood of 50%
(B) A Quarantine policy where agents were required to stay home when
symptomatic.
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Viral Strategies
Two strategies were investigated for the viral strain:
(A) An aggressive strategy in which the virus remains asymptomatic and
symptomatic for an equal period of days on average
Days after
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2
3
4
5
6
7
previous stage
Probability of
0.0
0.0
0.0
0.3
0.7
0.9
1.0
asymptomatic
period ending
Probability of
0.0
0.0
0.0
0.3
0.7
0.9
1.0
symptomatic
period ending
(B) A hiding strategy in which the virus remains asymptomatic for a period
two days longer on average than (A) and is symptomatic for two days
fewer.
Days after
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2
3
4
5
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7
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previous
stage
Probability of 0.0
0.0
0.0
0.0
0.0
0.3
0.7
0.9
1.0
asymptomatic
period ending
Probability of 0.0
0.3
0.7
0.9
1.0
----symptomatic
period ending
The graph above displays the most likely number of days that the virus will spend in
the asymptomatic and symptomatic phases of the SEiIR infection model based on
the strategy that the virus adopts.
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Payoff Calculations
Viral Payoff
The viral payoff was calculated as the sum of the number of individuals infected,
representing the virus’s reproductive success.
Public Health Payoff
Initially, it might seem as if the public health payoff clearly would be inverse to the
number of infections. This produces a zero-sum game, which, while simple, results
in obvious dominant strategies for both sides, meaning policy-makers do not need
to rely on game theory to determine whether to use a quarantine in such a situation.
A quarantine is always better in terms of minimizing infections than no quarantine.
Examining the economic effects as a payoff, however, produces more relevant
outcomes and interesting games.
(I)
Minimize Economic Impact – The public health payoff was defined so as to
value strategies that are best able to limit the economic effects of the
epidemic. Economic effects were determined by the effective work lost.
For the Quarantine strategy the effective work lost is evaluated simply by
the number of agents infected since this is directly proportional to the
number of symptomatic workers who were required to stay home.
For the Laissez-faire strategy the effective work lost is evaluated using the
equation:
æ
ì Effective ü
ç
ï
ï
í Work ý = - ç
ï Lost ï
ç
î
þ
è
ì# Infected
ï
í Workers
ï at Home
î
æ
ü
ì Infected ü
ï
ï
ï
ç
ý + ç 1- í Worker ý
ï
ï Productivity ï
ç
þ
î
þ
è
öì# Infected ü
ï
֕
÷í Workers ý
÷ï at Work ï
þ
øî
ö
÷
÷
÷
ø
The number of infected workers at home is directly proportional to half
the number of agents infected (because of the 0.5 probability that an
symptomatic agent will opt to stay home). The number of infected
workers at work is also evaluated as half the total number of infected
agents.
The infected worker productivity quantifies to what degree a worker can
complete their work duties while symptomatic. This parameter can vary
from 0 (no work accomplished) to 1 (no effect on productivity). Infected
worker productivity was varied in this research to examine how the
resulting Nash equilibria change depending upon its value.
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Data and Results
Simulation Results
Daily Infections by Public Health Strategy
(Viral Hiding Strategy, "Sniffles" Iterative
Dominance Game)
60000
Daily Infections
50000
40000
30000
Public Health Quarantine
Strategy
20000
Public Health Laissez-Faire
Strategy
10000
0
0 5 101520253035404550556065707580859095
Days Into Epidemic
Daily Infections by Viral Strategy (LaissezFaire Public Health Strategy, "Sniffles"
Iterative Dominance Game)
60000
Daily Infections
50000
40000
30000
Viral Aggressive Strategy
20000
Viral Hiding Strategy
10000
0
0 5 101520253035404550556065707580859095
Days Into Epidemic
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Two graphs of data selected from those produced by the simulations are displayed
above. Note that infections are higher for the viral Hiding strategy than for the viral
Aggressive Strategy, demonstrating that the virus has no incentive to deviate from
the viral Hiding strategy’s Nash Equilibrium.
Based off the earlier simulations and payoff calculations, two games were selected.
The Sniffles - An Iterative Dominance Game
The Iterative Dominance Game occurred in the case where workers had moderately
disabling symptoms and were moderately infectious when asymptomatic. The key
parameters in the simulations were:
Infected worker productivity = 50%.
Asymptomatic infectivity factor=50%
These conditions resulted in the following raw payoffs:
This game can be reduced for each player to a simpler format while maintaining the
value of each payoff relative to the other payoffs, in effect ranking each payoff in
terms of desirability for each player, with 4 being the most desirable and 1 the least
desirable.
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At first, the public health organization appears to have no clear dominant strategy,
with Quarantine being the better strategy if the virus adopts the Aggressive strategy
and Laissez-faire being the better strategy if the virus adopts the Hiding strategy.
However, by treating the virus as a rational player it is clear that the virus will
always gets a better outcome by adopting the Hiding strategy, and therefore can be
expected to choose this strategy. The public health organization should therefore
choose the Laissez-faire strategy and secure for itself the higher payoff. This results
in a Nash Equilibrium at (Laissez-faire, Hiding). If the public health organization
attempts to go for the most valuable payoff at (Quarantine, Aggressive) without
considering the virus’s role as a rational player, the organization will end up with its
least desirable payoff.
Spattergroit - A Prisoner’s Dilemma Game
A Prisoner’s Dilemma Game occurred in the case where workers had highly
disabling symptoms and were weakly infectious when asymptomatic. The key
parameters in the simulations were:
Infected worker productivity = 30%.
Asymptomatic infectivity factor=38%
These conditions resulted in the raw payoffs on the following page:
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The game can be reduced to a simpler format using the same ranking approach as
before.
For both the virus and the public health organizations, this game has a dominant
strategy, to quarantine and to hide, respectively, giving a Nash Equilibrium at
(Quarantine, Hiding). Note, however, the presence of the (Laissez-Faire, Aggressive)
strategy pair that results in higher payoffs for both players than the outcome at the
Nash equilibrium. This produces an interesting result in that the solution to the
Prisoner’s Dilemma (the classification of a game in which the Nash equilibrium is
not the most favorable outcome for the group) relies on either communication,
trust, changing the payoffs, or a repeated game. While the first two are impossible
with a virus, the third and fourth pose interesting sets of solutions. Through the
innate repetition of the game from outbreak to outbreak and the use of antivirals
and other means of slowing the spread of a virus, it could be possible to enter into a
cycle of cooperation with a virus in which each party chose strategies favorable to
both.
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Conclusion
Game Theory provides a useful tool for evaluating the interaction between
public health organizations and infectious disease. By considering infectious
diseases as rational players that make best strategy choices for their reproductive
success, effective public health strategies can be reached in situations that at first
might seem ambiguous. By considering economic costs in the game theoretic
analysis of these situations, non-zero sum games arise with interesting implications
for public health policy.
In the Sniffles scenario, game theory provided public health organizations
with a more clear strategy in a situation without a clear dominant strategy,
demonstrating the usefulness of this approach. In addition, the number of
microorganisms that have developed a symbiotic relationship with humans can
show this game’s relevance. The game’s result illustrates a possible cause for such
an occurrence in which microorganisms chose to increase their asymptomatic
period and decrease the severity of symptoms, while humans, cognizant of the
infection’s limited severity, chose not to restrict the microorganisms’ spread.
In the Spattergroit scenario, it was shown that the dominant strategies for
both the virus and human players resulted in an outcome favorable to neither, in a
reflection of the famous Prisoner’s Dilemma game. Games similar to the Prisoner’s
Dilemma have often been studied in order to devise a cooperative outcome better
for both players. It has been shown that in most cases a Prisoner’s Dilemma can be
solved either through communication between players, trust, changing the payoffs,
or repetition. While communication and trust are impossible to form with a virus,
the last two provide interesting methods to elicit cooperation (in the sense of both
players receiving a favorable outcome). Changing the payoffs might be accomplished
by using antivirals targeted against virus strains that failed to choose a cooperative
strategy, thereby encouraging the virus to choose an Aggressive strategy over
Hiding. In a repetitive game in which the number of repetitions is indefinite, there is
a decided advantage in choosing a cooperative strategy each round that the game is
played, if the other player also appears to be willing to cooperate. By combining
these two strategies, it is possible that health organizations could create cycles of
cooperation with viruses in which humans had minimal economic damages and
viruses had a chance to reproduce symptomatically, - a series of interactions that
could either be continued in a state of mutual cooperation or end in a sudden and
devastating defection by health organizations in order to eliminate the virus once
and for all.
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Future Research
Future research into using game theory in the fight against infectious disease should
initially include lab studies to determine the ability of viral organisms to react to a
human player. While the data showing that viruses and other organisms often
interact following game-theoretic principles among one another, games should still
be set up in laboratories in order to practically test the results shown here. In
addition, in the field of further simulations, it was posited that a game opposite to
the example shown with enterobacteriaceae could exist in which human
interference causes increases in the severity of bacterial symptoms, a potentially
interesting game. Should the methods be confirmed by the further study detailed,
game theory should begin to be applied to judge the proper response to epidemics
and infectious disease outbreaks and thereby minimize the threat that infectious
disease poses on society and maximize public health’s payoffs. By treating viruses as
rational players, the potential benefit of using game theory to develop strategies
that prevent epidemics could be enormous and critical to the success of public
health organizations worldwide.
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